To obtain images of the subsurface, a seismic method is often used, which consists in creating and sending seismic waves in the ground using sources such as explosives or vibrator trucks on land, or airguns offshore. The seismic waves penetrate the ground and get bounced, or reflected off geological discontinuities in the subsurface. As a result, they come back to the surface, where they are recorded using arrays of three component geophones (on land), or hydrophones (offshore) which are regularly distributed to cover areas of several square kilometers.
FIG. 1 illustrates diagrammatically a survey of seismic data with a source S of seismic waves and an array of receivers G. It also shows a point B of the subsurface which is assumed to contribute to the signal sensed by one of the receivers G. The horizontal coordinates of point B of the subsurface are denoted by x, y (or only one spatial coordinate if 2D imaging instead of 3D imaging is considered), while its depth is denoted by z. FIG. 1 also provides a simplified representation (dashed lines) of the propagation of seismic waves from the source S to the point B and from the point B to the receiver G. The waves are refracted at discontinuities of the geological layers where the acoustic impedance changes and reflected or diffracted at different positions including that of point B.
The data recorded in a seismic survey include, for each shot from a source S and for each receiver G, a seismic trace which is a time series of the signal sensed by the receiver G. The traces for a number of shots must be transformed to provide an image of the subsurface which will be the result of stacking or integrating a large amount of information. An important step of the transformation is the migration which consists in rearranging the data with respect to a model such that the stacking can be carried out coherently. The model is usually a map of the propagation velocity of the acoustic waves in the subsurface. It is not known a priori and it is a main challenge of all seismic imaging technologies to determine a model that will properly account for the field data after stacking.
In pre-stack depth migration (PSDM) methods, migrated data are computed for each shot using the velocity model and arranged in an output cube containing migrated values associated with positions in the subsurface. The cubes obtained for different shots are then analyzed to check consistency of the model. The model may be corrected and the process is iterated until a satisfactory image is obtained.
Common Image Gathers (CIGs) are popular tools for evaluating the migration velocity field and for imaging enhancement. They are made of data extracted from the output cubes, sorted in a convenient way for analysis so as to check the velocity model. A CIG is a bi-dimensional data structure defined for a given horizontal position x, y, with a first axis representing the depth z and a second axis representing a domain parameter A referred to for sorting the data of the output cubes. It contains reflectivity values obtained from the output cubes resulting from the migration, forming an image which can be analyzed to check and/or correct the velocity model. In this image, a pixel value at a point (z, A) represents a migrated value derived as a contribution of the subsurface position x, y, z to a seismic trace associated with the domain parameter A. Examples of commonly used domain parameters A include the surface offset, namely the distance between the source location for a shot and the receiver location providing the relevant trace for that shot, or the scattering angle at the subsurface position x, y, z.
The computation of common image gathers is not straightforward in all wavefield extrapolation methods. In fact, to date, depending on the propagation method employed to perform the migration, the output cubes are sorted in different ways. Even if there is no theoretical reason for this, the most practical way to produce common image gathers is selected according to different types of migration.
For example, scattering angle CIGs or surface offset CIGs are of widespread use in ray-based tomography techniques, to look for the velocity model which gives ‘flat gathers’. FIGS. 2A-C illustrate the principle in the case of surface offset CIGs. We consider a flat reflector at a position x0, y0, z0 of the subsurface. If the velocity model is correctly estimated, the reflector will provide a peak in the migrated signal at depth z0 for all values of the offset h, thus giving a flat event in the CIG as shown in FIG. 2A. If, however, the migration velocity is overestimated in the model, the same peak observed at the same time in a seismic trace will be associated with a larger depth value z1 as shown in FIG. 2B, and the corresponding event in the CIG will have a concave shape since the discrepancy in the wave travel time increases with the offset h. Likewise, if the migration velocity is underestimated in the model, the peak will be associated with a shallower value z2 as shown in FIG. 2C, and the corresponding event in the CIG will have a convex shape. If the reflector has a non-zero dip angle, its image in the CIG will be shifted horizontally in addition to vertically. Different tools have been developed to analyze the surface offset CIGs in order to correct the velocity model.
However, such tools have been used mostly in migration methods based on estimation of travel times between reflectors and the surface. More sophisticated migration methods have been developed to build PSDM images by solving the wave equation so as to obtain more accurate reflector amplitudes and structural positioning. For example, reverse-time migration (RTM) is a two-way migration solution which can accurately describe wave propagation in complex media. It is increasingly used in seismic exploration by virtue of advances in computer power and programming.
The above-mentioned analysis tools are not used with wave equation PSDM methods, including RTM, because it is not known how to compute surface offset CIGs.
In “Offset and angle-domain common image-point gathers for shot-profile migration”, Geophysics, Vol. 67, No. 3, 2002, pp. 883-889, J. Rickett and P. Sava established the notion of subsurface offset CIGs which requires the extension of the imaging condition through the computation of the correlation function along the spatial horizontal dimension. This type of gathers is the most common way to output wavefield-based migration images. It is better suited to techniques based on focusing analysis, which look for the highest correlation at zero-time lag and/or zero-offset. In “Angle-domain common image gathers by wavefield continuation methods”, Geophysics, Vol. 68, No. 3, 2003, pp. 1065-1074, P. Sava and S. Fomel proposed a method to derive scattering angle CIGs from subsurface offset CIGs. More recently, the same authors, by combining both time and depth, introduced the concept of extended image condition in “Time-shift imaging condition in seismic migration”, Geophysics, Vol. 71, No. 6, pp. 209-217.
Despite their benefits, some issues prevent the generalized usage of subsurface offset and angle gathers. Firstly, since RTM is a computer-intensive process, the computation of the cross-correlations for all the selected CIG locations adds a considerable extra cost. Secondly, subsurface offset gathers cannot be employed for classical tomography. The analysts cannot benefit from the full arsenal of tools developed for classical surface or angle CIGs (like RMO, Mute, AVO/AVA analysis, etc.). Subsurface scattering angle CIGs would be suited for this goal. However, these gathers imply an additional cost for a two-parameter Radon transform, and they do not show the correct kinematic move-out.
It would be desirable to obtain surface offset gathers with different kinds of migration methods, in particular wavefield methods including RTM, so as to keep the advantages of wavefield methods and, at the same time, address the limitation of the asymptotic assumption of ray-based methods, while sorting the migrated cubes in the same way as classical surface offset gathers.